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LESSONS IN PHYSIOLOGY.- No. IX.

MAN.

WE go on with our lesson on the nerves. You must keep in mind that the whole nervous system divides itself into distinct classes according to the functions or offices which the nerves have to perform. The following is the simplest arrangement which we can give :I. THE CEREBRO-SPI

NAL

NERVES, or those

which have their root in the spine and the brain, and serve the purpose of sensation and motion. By far the greater number of the nerves which supply the body generally, arise from the spinal marrow, and are possessed of double power. This double power is owing to the subdivision of the nerve into two parts or roots. One root passes to the posterior portion of the spinal cord, and has a ganglion on it. This bestows sensation alone. The other root goes to the anterior portion of that cord. And this gives motion alone.

II. THE EXCITO-MOTORY NERVES, or those

which pursue their course

from internal surfaces to the medulla oblongata or true spine, or that portion of the spinal marrow which in its ascent unites with the brain itself. These nerves are characterised by peculiar excitabilities; and when once excited pursue a reflex course from the medulla oblongata to the particular muscles on which they are designed to act,-the muscles concerned in the taking of food into the stomach, and then in throwing out whatever is not required

for the nutrition and building up of the body.

VI

III. THE RESPIRATORY NERVES, or those which pass off from this same medulla oblongata, and diverge to different parts of the head, neck, throat, and chest, as the divinely-prepared mechanism

Description of the Engraving.

1. Frontal branch of the fifth nerve of the brain which bestows sensation alone. 11. Superior maxillary, or that branch of the fifth nerve which supplies the upper jaw, and which, like the last, arising from the sensitive root, bestows sensation alone.

III. Mental or inferior maxillary branch of the fifth nerve. This also comes from the sensitive root. It is called mental, because it is involved in that expression which indicates the emotions of the mind.

IV. Temporal branches of the same fifth nerve. They are distributed on the temples, and are for sensation.

V. The only branch of the fifth nerve which arises from the smaller or motor root, and assists in the motion of those muscles which are employed in mastication or chewing.

VI. VII. VIII, IX. These are spinal nerves; the first of the series which come out between the vertebræ, in the whole length of the spine, to supply the body generally with motion and sensation.

A. The facial nerve. It is situated in the front of the ear, and is the VOL. I.

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Now as no animal can be compared with man in the number and range of his faculties, so there is not one which approaches him in the development and perfection of his nervous system. The connexion between this nervous system and the brain is very close; and not less close is the relation between the brain and the mind. The mind becomes conscious of external objects only by the influence which they exert on that part of the brain which is called THE SENSORIUM, from its being the peculiar seat of sensation. An impression is made upon any part of the surface of the body by mechanical contact, by heat, by electricity, or by any other external agent; this impression is conveyed by the nervous system to the sensorium, a certain change takes place in the condition of the brain, and the impression is then said to be felt, that is to say, the mind becomes conscious of it. Now it is not the impression made on some remote organ of the body by which the mind is influenced, but simply and immediately by the change which has taken place in the brain. This admits of easy proof. Let us sup

motor nerve of the features. It sends branches (a) to the muscles of the forehead and eyebrows. Branches (b) to the eyelids. Branches (e) to the muscles which move the nostrils and upper lip. Branches (d) to the lower lip. Branches (e) going down to the side of the neck. Connexions (with the spinal nerves of the neck. A nerve (g) to a portion of the muscle that is in the back of the head, and to muscles of the ear.

B. The nervus vagus, or the wandering nerve, so named from its extensive distribution. This is the grand respiratory nerve.

C. The spinal accessory nerve. D. The ninth nerve, which is the motor nerve of the tongue. E. The nerve which supplies the diaphragm. F. Branch of the sympathetic nerve.

G. A branch of the nervus vagus, which goes to the superior portion of the larynx or windpipe.

H. Another branch of the vagus, which goes to the inferior portion of the larynx. I. The nerve which goes to the tongue and upper part of the gullet called the pharynx.

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pose, as is really the case, that the mind remains perfectly capable sensations. It is common to nearly the whole surface of the body; of receiving any impression, but that all communication with the but nowhere is the sensibility so keen or acute as at the end of the brain is cut off. What is the consequence? Let the pressure fingers. The following cut shows the distribution of the nerves of which is applied to any distant part of the nervous system be what touch at the extremity of the thumb; it may, no impression is felt, we have no consciousness of it. For and the human hand, both by reason of example:-"The surface of the eye-a part so exquisitely sensible, being endowed with a larger supply of that if a fine hair touches it, there will be severe pain and spasm of sensitive fibres, and of its peculiar comthe eyelids-may, when the nerve is destroyed, be rudely pressed formation, is, par excellence, the organ of with the finger, and the patient will nevertheless be unconscious of touch. But the sense of touch in the pain; or, if the surface be inflamed, and it be necessary to scarify hand would have been of little use withit with the point of the lancet, in order to withdraw blood, the out the power of motion. It is in that patient will submit to the operation without pain, and without even wondrous variety of movement of which winking, although the eyelids retain their power of closing." It the hand of man is capable, that we see follows from this, that in the process by which the mind is rendered his physical superiority to every other conscious of external objects, there are three distinct steps or animal. Remember, moreover, that stages:while the greatest sensibility is felt at the extremities of the fingers and in the lips, the least sensibility is in the skin of the trunk, the arm, and the thigh. Besides, it is a question, whether the sense of temperature is not conveyed by a set of fibres altogether distinct from those which minister to the proper sense of

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:

i. The reception of the impression which is made at the extremity of the nerve of sensation.

ii. The conducting of this impression along the trunk of the nerve to the sensorium or the brain.

iii. The change excited by this impression in the sensorium itself, through which sensation is produced.

The sensitive extremities of the cerebro-spinal nerves transmit all their impressions to the brain under the form of sensations, "Although we commonly refer our various sensations to the parts at which the impressions are made; as, for instance, when we say that we have a pain in the hand, or an ache in the leg, we really use incorrect language; for though we may refer our sensations to the parts where the impression is first made on the nerves, they are really felt in the brain. This is evident from two facts :-first, that if the nervous communication between the part and the brain be interrupted, no impressions, however violent, can make themselves felt; and, second, that if the trunk of the nerve be irritated or pinched anywhere in its course, the pain which is felt is referred, not to the point injured, but to the surface to which these nerves are distributed. Hence the well-known fact, that for some time after the amputation of a limb, the patient feels pains which he refers to the fingers or toes that have been removed. This continues until the irritation of the cut extremities of the nervous trunks has subsided."

touch or resistance.

TASTE. This sense has for its organ that portion of the nervous which is distributed to this membrane, terminates in little bodies or membrane which covers the tongue and the throat. The nerve accumulations called papillæ, or clusters of papillæ, as you see in the cut, which represents the fungus-like papilla of the tongue.

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Its chief purpose is to direct animals in their choice of food, and hence the wisdom of placing its organ at the entrance to the digestive canal. The sense of taste is of a much more refined nature than that of touch; but, like that of touch, is excited by direct contact with particular substances. This contact produces either an agreeable or a disagreeable sensation. The savour may be strong, or slight, or insipid, and the impression will correspond. But the impression produced by every substance taken into the mouth, depends as much on smell as on taste. If you take some decidedly aromatic body into the mouth, and press the nostrils close together, you will scarcely be conscious of any taste. Withdraw the pressure from the nostrils, and leave their passage perfectly free, and immediately you become conscious of a certain impression or sensation.

But while there is this common sensibility which is diffused over the greater part of the body, there are organs of special sense, endowed with the property of receiving impressions peculiar to themselves. These senses are five-touch, taste, smell, sight, and hearing-while the impressions derived from them not only differ from all our other sensations, but differ the one from the other. The sense of touch would never give us the idea or knowledge of those properties which are discovered by taste; nor would the idea of form, shape, or bulk, ever be derived from the sense of smell. The eye would never distinguish sounds, nor would the ear ever | discriminate colours. Who would ever think of saying that he hears a beautiful flower, or sees an enchanting sound, or smells a square body, or grasps a disagreeable odour? Each sense has its own nerve, and though "we can acquire a knowledge of the shape and position of objects by the touch, we could form no notion of their colour without sight, of their sounds without hearing, or of their odours without smell." Have, then, these nerves of special sense nothing in common? Are they not all susceptible of impressions inseparable from a feeling of pain? It cannot be denied that any violent or excessive impression may occasion a certain degree of discomfort. For example:-The effect of strong daz-extreme minuteness. These particles proceeding or flying off from zling light, or of any sudden unexpected sound, or of any powerful odour, or of any very decided taste, would be followed by corresponding uneasiness. But mark, this is only a question of degree. These very sensations, in a more or less degree, might afford a certain pleasure or delight, or even in the same degree in a different condition of the body. Our feeling of pleasure or of pain very much depends on the previous condition of the part affected, as also upon the extent of surface on which an impression is made. For example:-If you put one hand in hot water, and the other in cold, and then transfer both to tepid water, this tepid water will seem cool to the one, and warm to the other. The second fact may be proved thus:-Hot water, into which you might introduce a single HEARING.-The ear presents a very beautiful, but very complifinger without any unpleasant sensation, would, on plunging the cated apparatus. With the form and the situation of the external whole hand into it, not only scald the hand, but give you the feel-ear you are already familiar. The apparatus more properly coning of pain. sists of the tympanum or drum, with its membrane and chain of TOUCH is the most general and the most diffused of our external bones; and which is adapted to direct itself, more or less com

SMELL. If the thousand varieties of savour, may, in relation to the sense of taste, be all resolved into agreeable and disagreeable, then analogous characters pertain to those various odours, which, in relation to the sense of smeil, are either pleasant or repugnant. This sense of smell depends upon the diffusion of the particles of the substance through the surrounding air, in a state of larger bodies, take on a certain degree of volatility, and coming into contact with the mucous membrane which lines the nasal cavities, and in which the olfactory nerve is distributed, we become conscious of a corresponding sensation. The atmospheric air, in its passage through the nostrils, is, during inspiration, the vehicle of those odorous particles into the olfactory organ, and by coming into immediate contact with its mucous membrane, informs the animal of the presence, at a greater or less distance, of the bodies which are the source of these odours. In proportion to the extent of this membraneous surface, is the acuteness of the sense. In this respect man is far surpassed by many of the lower animals.

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that membrane. Beyond the membrane is this auditory cavity, which is filled with the air which it receives through the Eustachian tube; and behind this cavity, and opposite to the membrane of the tympanum, is a membrane of the same kind with two apertures, the one oval, and the other round, and these two membranes are united by a chain of little bones. The vibrations produced in the membrane of the tympanum, are transmitted to the membrane with the oval aperture by means of this little chain of bones, from the air contained within the osseous walls of the cavity. This cavity, again, is in communication with the vestibule, which borders upon a system of semicircular canals, which together form the labyrinth or internal ear. These canals contain a watery liquid, into which the fibres of the auditory nerve plunge. The vibrations of the membrane with its apertures are transmitted hither; and the nerve conveying these impressions to the brain, we have the sensation of hearing. SIGHT.-The eye, as the organ of vision, is made up of an assemblage of parts even more complex than those of the ear. an optical instrument of wonderful completeness. This beautiful instrument presents the form and appearance of a globe, and consists of a spheroidal envelope-THE SCLEROTIC-which offers to its anterior part, a transparent circular segment-THE CORNEA-for Crystalline lens.

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the passage of the rays of light, and to its posterior portion, a round opening for the introduction of the optic nerve, which at once developes itself in a nervous membrane-THE RETINA-doubling within the sclerotic in a manner to curtain three-fourths of the posterior of the spheroidal cavity of the globe. It is upon this nervous expansion that the luminous rays fall, and penetrating through it, terminate upon an absorbent membrane, impregnated with a black matter. This membrane is named THE CHOROID, and is that coat which is found immediately under the sclerotic. At the level of the union of the cornea with the sclerotic, the cavity of the eye is separated into two parts by a moveable, vertical, and diversely-coloured partition-THE IRIS-which is pierced in the centre by a round opening-THE PUPIL. The anterior parts of the cavity of the eye is filled by a transparent liquid-THE AQUEOUS HUMOUR. The posterior part contains THE CRYSTALLINE LENS; situated behind the iris, enclosed in a concavity of the vitreous body, which is a transparent mass made up of little cells full of watery liquid, and this liquid fills all the posterior cavity, even to immediate contact with the retina.

Through the various transparent media of the cornea, the aque

ous humour, the crystalline lens, and the vitreous body, theys of light which diverge from the several points of any object, must pass to reach the retina. Falling upon the front of the cornea, they are refracted by its convex surface, while passing through it into the eye. Here they slightly converge; but are brought more closely together by the crystalline lens, which they reach after passing through the pupil. The refracting influence of the lens, together with that produced by the aqueous humour, is such as to bring these rays to a focus on the retina, where is formed a complete inverted image of the object. Then to prevent the light that forms the picture from being reflected from one part of the interior of the globe to another, which would mar and confuse the picture, the retina, which is almost transparent, is spread over the layer of black pigment which lines the choroid coat, and is designed to absorb the rays as soon as they have passed through the retina. The impression of the image is then transmitted by the optic nerve to the brain, and we become conscious of a certain sensation. According to Sir Charles Bell's theory and arrangement, the distribution of the nerves to the face and neck is seen in the figure p. 353. An arrangement this which again proclaims the infinite skill and love of Him whose hands fashioned our wondrous frame, and in whom we live, and move, and have our being.

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Which are the organs of special sense?

Which is the grand distinguishing organ of touch in man? Describe how we get our impression of objects by the sense of sight.

How are the rays of light kept from being reflected from one part of the globe of the eye to another?

Where are the nerves of voice and expression distributed?
To whom are we indebted for this discovery?

and perfection of his nervous system?
Is there any animal which approaches man in the development

What feelings should this awaken in us towards the great Creator?

LESSONS IN LATIN.-No. XXI.
By JOHN R. BEARD, D.D.
REGULAR VERBS.

THE SECOND CONJUGATION.

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Indicative. monui

II. FUTURE.

Plural.

PLUPERFECT.

Sing. Plural.

PERFECT.

Sing. Plural.

Sing.

Subjunctive.
monuerim
monueris

monuisti

monuit

monúerit

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monuerunt monuerint

monúeram

monúeras
monterat

monueramus
monuerátis
monuerant

montero
monúeris
monúerit

monuerimus
monuéritis
monuerint

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He reminded me; they reminded the king; I might remind you; you might remind me; they have reminded the boy; thou wast reminding the woman; I will remind the teacher; be silent; do you be silent; let them be silent; the woman was suddenly silent; take care to improve; that thou improvest the morals of the citizens; I fear he may displease thee; the boys feared to displease their father; he pleases all (persons); a good man will displease the bad; why art thou silent? they fear that Caesar will conquer their country; good sisters fear (their) brothers will not be well; art thou well? I fear thou wilt not be well; if thou exercisest thy body thou wilt be well; my mother fears an entrance into heaven will (may) not lie open to me.

In reply to some inquiring correspondents, the following information is added to the subject of nouns :

The names of the cases in Latin are of Greek origin, as is the term case (ptosis, fall) itself. The nominative is so called because it assigns the name (nomen) of the agent; the genitive denotes the relation of genus, origin or birth, that out of which a thing arises, and to which therefore it belongs; the accusa

EXERCISES:-Like moneo form doceo, I teach; and habeo, tive points out that which is caused (causa) or effected, that

I have.

VOCABULARY.

Exerceo 2, I exercise; gaudeo 2, I rejoice; gaudeo quod, I rejoice that; valeo 2, I am well; pareo 2, 1 obey; placeo 2, I please; displiceo 2, 1 displease; oblivio, ónis, f. forgetfulness; deleo 2, I blot out; floreo 2, I flourish; probe, honestly, properly; taceo 5, I am silent (E. R. tacit); repente, adv. suddenly; aditus, ûs, m. access, entrance (E. R. an adit); pateo 2, I lie open, I am open; timeo 2, I fear (E. R. timid); noceo 2, I injure; venio 4, I come; occido 3, I fall, die.

Observe that, occasionally, forms of other conjugations, &c., than the one immediately under treatment, are introduced with a view to keep up the student's attention, exercise his ingenuity, and test his progress.

RULE.-After verbs expressive of fear, ne is used with the subjunctive of the following verb. Ne is, in Latin, a negative, and may be often rendered by lest; it may, however, in construing into English, be altogether dropped, the ordinary connecting particle that, or that not, being put between the two verbs; e. g.,

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which receives the action implied in the verb, the result of that action, the object or thing on which the action falls; the dative case is the giving (do, dare, dedi, datum, to give) case, signifying the person (or thing) to which something is given or assigned; the vocative (voco, I call) is the case of calling on or addressing; and the ablative (ablatus, taken from) bears the name because it involves the idea of separation. In the Greek there is no ablative, the relations implied in that case being expressed partly by the genitive, partly by the dative.

LESSONS IN GEOMETRY.-No. X.
LECTURES ON EUCLID.

DEFINITIONS. BOOK I. FROM VIII. TO XII. INCLUSIVE.

VIII.

[A plane angle is the inclination of two lines to one another in a plane, which meet together, but are not in the same direction.] This definition is put in brackets, as useless, and unnecessary to be remembered.

IX.

A plane rectilineal angle is the inclination of two straight lines to one another, which meet together, but are not in the same straight line. When two straight lines meet at a point, so that if produced they would intersect (cross) each other, the indefinite space between them is called an angle.

N.B. "When several angles are at one point B, any one of them is expressed by three letters, of which the letter that is at the vertex of the angle, that is, at the point in which the straight lines that contain the angle meet one another, is put between the other two letters, and an one of these two is somewhere upon

Hence, you see, that in putting similar sentences into English, you must drop the negative where it is found in the Latin, and use it where it is not found in the Latin.

EXERCISES.-LATIN-ENGLISH.

Exerceo; exercebam; exercebat; exercebo; gaudeo quod tu vales; præceptor gaudebat quod vos ejus præceptis parebatis; tibi placebas, aliis displicebas; virtutis honorem nulla oblivio delebit; exercai; Græcia omnibus artibus floruit; laudo vos quod mentes vestras studio probe exercuistis; cur tacuistis? Tacuit puer reperte; tacebat mater; tacent omnes; nisi virtutis præceptis, parueritis, aditus in coelum vobis non patebit; si cupiditates tuas coercueris, beatus eris; curo ut pueri mores emendem et corpus exerceam; moneo vos ut patris praecepta observetis; timebam ne vobis displicerem; cura ut pueri mores emendes et corpus exer

B

D

one of those straight lines, and the
other upon the other straight line:
thus the angle which is contained by
the straight lines, A B, C B, is named
named the angle A B D, or D BA; and that which is contained by
the angle ABC, or C BA; that which is contained by AB, D B, is
DB, CB, is called the angle DB C, or C BD; but, if there be only one
angle at a point, it may be expressed by a letter placed at that point;
as the angle at E." This explanation is put in inverted commas, as
being Dr. Simson's addition; it is very necessary to be remembered.
X.

When a straight line standing on another straight line
makes the adjacent angles equal to one another, each of these
angles is called a right angle; and the straight line which
stands on the other is called a perpendicular to it. By bor-
rowing the terms vertical and horizontal from the lan-
guage of Physics, we may define a right angle to be that
which is formed by the meeting of a vertical and horizon-
tal line.

L

XI.

An obtuse angle is that which is greater than a right angle.
XII.

An acute angle is that which is less than a right angle. The term oblique angles is applied both to obtuse angles and to acute angles.

On the eighth definition, Dr. Simson remarks that it seems he who made it, intended that it should comprehend not only a plane angle contained by two straight lines, but likewise the angle which some conceive to be made by u straight line and a curve, or by two curve lines which meet one another in a plane. Now, though the words, in the same direction, are easily understood when applied to two straight lines, it does not appear clearly what ought to be understood by them, when applied to a straight line and a curve, or to two curve lines. Besides, all the angles that are considered and treated of in the Elements of Geometry, are those made by straight lines only; and, therefore, the definition and consideration of such other angles as are above described, is wholly unnecessary and uncalled for. It may be further remarked, that as the doctrine of curves belongs to the Higher Geometry, sometimes called Transcendental, it is plain that the consideration of curvilinear or mixtilinear angles must be deferred until the student has made himself master of the Elements of Euclid. Dr. Simson seems to imply, in the preceding observation, that this eighth definition is not Euclid's, although found in the Greek text; indeed, he asserts, in another place, that it is the addition of "some less skilful editor;" by this, he evidently means an editor less skilful than the author; but wherever any error is found in the Greek text, Dr. Simson never admits that it is Euclid's error, for Euclid is his hero; but always throws the blame upon his editors.

The ninth definition is one of very great importance to the student, as upon the right understanding of what an angle really is, all his future progress in Geometry depends. Various modes have been proposed in order to convey the idea of an angle clearly to the mind by words; but, at first, it is always surrounded with a haze of indistinctness, whatever words may be employed. An angle (or a rectilineal angle) is sometimes said to be the degree of opening or divergence of two straight lines which meet one another. This definition plainly implies that the lines have been at first shut together, or coincident with each other, and that they have been separated at one end, while they were kept together at the other. Hence, Dr. Thomson throws himself at once upon this idea, and says, "a clear idea of the nature of an angle is obtained by gradually opening a carpenter's rule or a pair of compasses; as the angle made by the parts of the rule or the legs of the compasses, will become greater as the opening widens." This illustration will do very well for a class, if the members, while listening to the lecturer, do not imitate the operation of the compasses with their mouths, that is, in plain English, do not yawn over it, as they are very apt to do. In an edition of Euclid's Elements, formerly referred to, and which we shall in future call the Gower-street edition, the editor plunges at once into the idea of an angle generated by motion, and endeavours to enlarge our conception of an angle to a degree far beyond that contemplated by Euclid. He very truly says that angles might not improperly be considered as a fourth species of magnitude;" the other three species being lines, surfaces, and solids. Angular magnitude evidently consists of parts, and must therefore be admitted to be a species of quantity. The student must not suppose that the magnitude of an angle is affected by," or depends upon "the length of the straight lines which include it," or by which it is formed, "and of whose mutual divergence it is the measure. These" straight lines, which are called the sides or the legs of the angle, are supposed to be of indefinite length. To illustrate the nature of angular magnitude, we shall recur to motion." This illustration, which is given by many editors, consists in supposing a straight line to be fixed at one end, and extended indefinitely in one given direction; it is then supposed to leave this initial, or first position, and to revolve round its fixed extremity in the same plane; as it revolves, the indefinite space between its initial position and every

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successive position, constitutes an angle; and this angle in-
creases in magnitude from zero, or nothing, to an entire
revolution; it may also be stopped at any point in the course
of this revolution, it may be carried to any point beyond this
revolution, or it may be made to perform any number of
revolutions from the same point, or from one point to another,
and in all these cases it will generate what is called an angle.
Hence, an angle may consist of any part of such revolution, a
whole revolution, a whole revolution and part of a revolution,
any number of revolutions, or any number of revolutions and
a part of a revolution. This idea of an angle was certainly
not contemplated in the Elements of Euclid.
It is even
asserted that no angle greater than half of such a revolution
was considered or contemplated by him. To this, we demur;
for we think there are traces in the third book of his
having taken into consideration angles which are greater than
those formed by half a revolution, as well as those which are
formed by less than half a revolution. With the idea of
revolution in our hands, we can give very easy explanations
of some subsequent definitions: thus, a right angle is one-
fourth of an entire revolution; an acute angle is less, and an
obtuse angle greater than this quantity. Two right angles are
half of an entire revolution; three right angles, three-fourths of
an entire revolution; and four right angles, an entire revolution.
Useful, however, as the idea of revolution is, in thus ex-
plaining the nature of angles, it is foreign to geometry, and
not strictly admissible into a logical treatise. The reason of
this is evident; it is only physical or natural objects that can
be put in motion; and therefore motion belongs to Natural
Philosophy and not to Geometry. A straight line is not a
physical object, but a mental or ideal object, and therefore it
cannot be put in motion; it may nevertheless be conceived to
be put in motion; but this would require a new postulate in
geometry; and this postulate being admitted, new postulates
would be required at every step, in order to carry on the rea-
soning. Euclid foresaw this, and therefore excluded motion
as much as possible from his Elements. It is true that the
idea of revolution occurs in the definitions of the twelfth Book;
but there its introduction is of less importance, seeing that he
has so firmly established the principles of geometry in the
preceding books; and it is admitted that the first six books
are more logically reasoned and elegantly demonstrated than
the subsequent books.

We cannot avoid calling the attention of the student most particularly to the N.B. or Nota Bene (that is, "mark well") added by Dr. Simson. It is plain that when two straight lines drawn in different directions, meet at any point in a plane they form, according to the usual interpretation of Euclid's definition, only one angle at that point; and therefore only one letter is necessary to be placed at that point in order to enable us to speak of that angle or to describe it. But when three straight lines drawn in different directions meet at any one point in a plane, they form at least three different angles, according to the usual interpretation of Euclid's definition; and, therefore, more letters than one are necessary to distinguish these three angles from one another. Now, if besides the point where the two straight lines forming any angle meet, which is called the vertex of the angle, two other points be taken, one on each of these straight lines or legs of the angle, at any convenient distance from the vertex, and letters be put at these three points, we shall be enabled to speak of that one angle individually without confusing the mind with the other two angles. And in order that the nature of the angular space may be clearly presented to the mind's eye by the language employed, the letter at the vertex of the angle is always placed in the middle between the other two letters, and the three letters are then read in succesion, not as a word, but as letters of the alphabet. Thus, in the diagram above, the three straight lines A B (a-bee), DB (dee-bee), and CB (see-bee) meet in the same point or common vertex B, and form the three angles A B D (a-bes-dec), DBC (dee-bee-see), and ABC (a-bee-see); or, which is the same thing, the three angles DBA (dee-bee-a), CBD (see-bee-dee), and CBA (see-bee-a); where the B, the letter at the vertex in each angle, is carefully preserved in the middle between the other two letters; so that it is quite immaterial to the sense, whether you begin to name an angle by the letter on the one leg, or the letter on the other leg, provided you keep the letter at the vertex in the middle between them.

The utility

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